QANOVA {GFD}

QANOVA: Quantile-based analyis-of-variance

Description

The function qanova calculates the Wald-type statistic based on the quantiles and/or their linear combinations, e.g. the interquartile range. Respective p-values are obtained by a \chi^2-approximation and a permutation approach, respectively.

Usage

QANOVA(formula, data = NULL, quantiles = c(0.5), lin_mat = NULL,
  var_method = "interval", nperm = 1999, var_level = 0.95,
  nested.levels.unique = FALSE)

Arguments

formula

A model formula object. The left hand side contains the response variable and the right hand side contains the factor variables of interest. An interaction term must be specified.

data

A data.frame, list or environment containing the variables in formula and the censoring status indicator. Default option is NULL.

quantiles

A vector of probabilties corresponding to the quantiles of interest. By default is c(0.5), i.e. just the median is included.

lin_mat

A matrix specifying which linear combination of the quantiles should be included for the analysis. By default (NULL) the identity matrix is chosen, i.e. all chosen quantiles are considered simulatenously.

var_method

Method for the variance estimation of the sample quantiles. The default ("interval") is the interval-based estimator of Price and Bonett (2001). Additionally, the bootstrap method ("boot") of Efron (1987) or a kernel density approach ("kernel") can be chosen.

nperm

The number of permutations used for calculating the permuted p-value. The default option is 1999.

var_level

A number between 0 and 1 specifying the confidence level for the interval variance estimation method; the default value is 0.95.

nested.levels.unique

A logical specifying whether the levels of the nested factor(s) are labeled uniquely or not. Default is FALSE, i.e., the levels of the nested factor are the same for each level of the main factor.

Details

The qanova function calculates the Wald-type statistic based on quantiles and linear combinations of them for general factorial designs. The procedure is fully nonparametric and no specific assumption of the underlying distribution is required. In particular, heteroscedastic settings can be studied. The analysis can be based on a single quantile (e.g. the median, default choice), a linear combination of quantiles (e.g. the interquartile range, set quantiles=c(0.25,0.75) and lin_mat = matrix(c(-1,1),ncol=2)) or on several (combinations of) quantiles simulatenously.

The qanova function returns the test statistic as well as two corresponding p-values: the first is based on a chi^2 approximation and the second one is based on a permutation procedure.

@return A qanova object containing the following components:

• pvalues_statThe p-values obtained by \chi^2-approximation

• pvalues_perThe p-values of the permutation approach

• statisticsThe value of the qanova along with degrees of freedom of the central chi-square distribution and p-value, as well as the p-value of the permutation procedure.

• npermThe number of permutations used for calculating the permuted p-value.

Author(s)

Philipp Steinhauer

References

Ditzhaus, M., Fried, R. and Pauly, M. (2021). QANOVA: Quantile-based Permutation Methods For General Factorial Designs. TEST (to appear, ArXiv preprint arXiv:1912.09146). Efron, B. (1979). Bootstrap methods: Another look at the jackknife. Ann. Statist., 7:1-26. Price, R. and Bonett, D. (2001). Estimating the variance of the sample median. J. Stat. Comput. Simul, 68:295-305.

Examples

QANOVA(weightgain ~ source*type, data = HSAUR::weightgain,var_method = "interval", nperm =199)